What is the integral of 90x^2sin(2+6x^3) ?

The integral of 90x^2sin(2+6x^3) is -5cos(2+6x^3)+C

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integral of 90x^2sin(2+6x^3)

\int 90 x^{2} sin (2 + 6 x^{3}) d x

- 5 cos (2 + 6 x^{3}) + C

Solution steps

\int 90 x^{2} sin (2 + 6 x^{3}) d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 90 \cdot \int x^{2} sin (2 + 6 x^{3}) d x

Apply u-substitution

= 90 \cdot \int \frac{sin ( 2 + 6 u )}{3} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 90 \cdot \frac{1}{3} \cdot \int sin (2 + 6 u) d u

Apply u-substitution

= 90 \cdot \frac{1}{3} \cdot \int sin (v) \frac{1}{6} d v

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 90 \cdot \frac{1}{3} \cdot \frac{1}{6} \cdot \int sin (v) d v

Use the common integral:

\int sin (v) d v = - cos (v)

= 90 \cdot \frac{1}{3} \cdot \frac{1}{6} (- cos (v))

Substitute back

= 90 \cdot \frac{1}{3} \cdot \frac{1}{6} (- cos (2 + 6 x^{3}))

Simplify

90 \cdot \frac{1}{3} \cdot \frac{1}{6} (- cos (2 + 6 x^{3})) : - 5 cos (2 + 6 x^{3})

= - 5 cos (2 + 6 x^{3})

Add a constant to the solution

= - 5 cos (2 + 6 x^{3}) + C

What is the integral of 90x^2sin(2+6x^3) ?
The integral of 90x^2sin(2+6x^3) is -5cos(2+6x^3)+C